It was undoubtedly a necessary task to collect
all the results on the concentration of measure during the past years
in a monograph. The author did this very successfully and the book is
an important contribution to the topic. It will surely influence
further research in this area considerably. The book is very well
written, and it was a great pleasure for the reviewer to read
it.
—Mathematical Reviews
The observation of the concentration of measure
phenomenon is inspired by isoperimetric inequalities. A familiar
example is the way the uniform measure on the standard sphere
$S^n$ becomes concentrated around the equator as the
dimension gets large. This property may be interpreted in terms of
functions on the sphere with small oscillations, an idea going back to
Lévy. The phenomenon also occurs in probability, as a version
of the law of large numbers, due to Emile Borel. This book offers the
basic techniques and examples of the concentration of measure
phenomenon. The concentration of measure phenomenon was put forward in
the early seventies by V. Milman in the asymptotic geometry of Banach
spaces. It is of powerful interest in applications in various areas,
such as geometry, functional analysis and infinite-dimensional
integration, discrete mathematics and complexity theory, and
probability theory. Particular emphasis is on geometric, functional,
and probabilistic tools to reach and describe measure concentration in
a number of settings.
The book presents concentration functions and inequalities, isoperimetric and
functional examples, spectrum and topological applications, product measures,
entropic and transportation methods, as well as aspects of M. Talagrand's deep
investigation of concentration in product spaces and its application in
discrete mathematics and probability theory, supremum of Gaussian and empirical
processes, spin glass, random matrices, etc. Prerequisites are a basic
background in measure theory, functional analysis, and probability theory.
Readership
Graduate students and research mathematicians
interested in measure and integration, functional analysis, convex and
discrete geometry, and probability theory and stochastic
processes.